Consider the situation shown in figure.
Question: Consider the situation shown in figure. The wire $\mathrm{PQ}$ has mass $\mathrm{m}$, resistance $\mathrm{r}$ and can slide on the smooth, horizontal parallel rails separated by a distance $\mathrm{I}$. The resistance of the rails is negligible. A uniform magnetic field $B$ exists in the rectangular region and a resistance $\mathrm{R}$ connects the rails outside the field region. At $\mathrm{t}=0$, the wire $\mathrm{PQ}$ is pushed towards right with the speed $^{v_{0}}$. Find (a) the c...
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Question: Find the point on the curve $y=x^{2}+3 x+4$ at which the tangent passes through the origin. Solution:...
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Question: Consider the situation of the previous problem (a) Calculate the force needed to keep the sliding wire moving with a constant velocity $v .$ (b) If the force needed just after $\mathrm{t}=0$ is $F_{0}$, find the time at which the time at which the force needed will be $F_{0} / 2$. Solution:...
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Question: Prove the tangent to the curve $y=x^{2}-5 x+6$ at the point $(2,0)$ and $(3,0)$ are at right angles. Solution:...
Read More →Figure shows a long U-shaped wire of length I placed in a perpendicular magnetic field
Question: Figure shows a long U-shaped wire of length I placed in a perpendicular magnetic field B. A wire of length I is slid on the Ushaped wire with a constant velocity $v$ towards right. The resistance of all the wires is r per unit length. At $t=0$, the sliding wire is close to the left edge of the U-shaped wire. Draw an equivalent circuit diagram, showing the induced emf as a battery. Calculate the current in the circuit. Solution:...
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Question: Find the point on the curve $x^{2}+y^{2}-2 x-3=0$ where the tangent is parallel to the $x$-axis. Solution:...
Read More →Figure shows a wire sliding on two parallel,
Question: Figure shows a wire sliding on two parallel, conducting rails placed at a separation I. A magnetic field B exists in a direction perpendicular to the plane of the rails. What force is necessary to keep the wire moving at a constant velocity v? Solution:...
Read More →A copper circular-ring of radius r translates in its plane with
Question: A copper circular-ring of radius $r$ translates in its plane with a constant velocity $v$. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the ring. Consider different pairs of diametrically opposite points on the ring. (a) Between which pair of points is the emf maximum? What is the value of this maximum emf? (b) Between which pair of points is the emf minimum? What is the value of this minimum emf? Solution:...
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Question: At what point on the curve $x^{2}+y^{2}-2 x-4 y+1=0$, is the tangent parallel to the $y$-axis? Solution:...
Read More →A wire of length 10 cm translates in a direction making and
Question: A wire of length $10 \mathrm{~cm}$ translates in a direction making and angle of $60^{\circ}$ with its length. The plane of motion is perpendicular to a uniform magnetic field of $1.0 \mathrm{~T}$ that exists in the space. Find the emf induced between the ends of the rod if the speed of translation is $20 \mathrm{~cm} / \mathrm{s}$." Solution:...
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Question: Find the equation of the tangent to the curve $x^{2}+3 y=3$, where is parallel to the line $y-4 x+5=0$. Solution:...
Read More →A copper wire bent in the shape of a semicircle of radius r
Question: A copper wire bent in the shape of a semicircle of radius $r$ translates in its plane with a constant velocity v. A uniform magnetic field $B$ exists in the direction perpendicular to the plane of wire. Find the emf induced between the ends of the wire if (a) the velocity is perpendicular to the dimeter joining free ends, (b) the velocity is parallel to this diameter. Solution:...
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Question: Show that the tangents to the curve $y=2 x^{3}-4$ at the point $x=2$ and $x=-2$ are parallel. Solution:...
Read More →A right-angled triangle abc,
Question: A right-angled triangle abc, made from a metallic wire, moves at a uniform speed $v$ as it is shown in figure. A uniform magnetic field B exists in the perpendicular direction. Find the emf induced (a) in the loop abc, (b) in the segment bc, (c) in the segment ac and (d) in the segment $a b$. Solution:...
Read More →The two rails of a railway track,
Question: The two rails of a railway track, insulated from each other and from the ground, are connected to a millivolt meter. What will be the reading of the millivolt meter when a train travels on the track at a speed of $180 \mathrm{~km} / \mathrm{hr}$ ? The vertical component of the earth's magnetic field is $0.2 \times 10^{-4} \mathrm{~T}$ and the rails are separated by $1 \mathrm{~m}$. Solution:...
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Question: Find the equation of the normal to the curve $y=(\sin 2 x+\cot x+2)^{2}$ at $x=\frac{\pi}{2}$ Solution:...
Read More →A 10 m wide spacecraft moves through the interstellar space at
Question: A $10 \mathrm{~m}$ wide spacecraft moves through the interstellar space at a speed $3 \times 10^{7} \mathrm{~m} / \mathrm{s}$. A magnetic field $B=3 \times 10^{-10} \mathrm{~T}$ exists in the space in a direction perpendicular to the plane of motion. Treating the spacecraft as a conductor, calculate the emf induced across ita width. Solution:...
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Question: Find the equation of the tangent to the curve $y=\left(\sec ^{4} x-\tan ^{4} x\right)$ at $x=\frac{\pi}{3}$. Solution:...
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Question: Show that the equation of the tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ at $\left(x_{1}, y_{1}\right)$ is $\frac{x_{1}}{a^{2}}-\frac{y y_{1}}{b^{2}}=1$. Solution:...
Read More →A metallic meter stick moves with a velocity of
Question: A metallic meter stick moves with a velocity of $2 \mathrm{~m} / \mathrm{s}$ in a direction perpendicular to its length and perpendicular to a uniform magnetic field of magnitude $0.2 \mathrm{~T}$. Find the emf induced between the ends of the stick. Solution:...
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Question: Figure shows a circular wheel of radius $10.0 \mathrm{~cm}$ whose upper half, shown dark in figure, is made of iron and the lower half of wood. The two junctions are joined by an iron rod. A uniform magnetic field B of magnitude $2.00 \times 10^{-4}$ T exists in the space above the central line as suggested by the figure. The wheel is set into pure rolling on the horizontal surface. If it takes $2.00$ seconds for the iron part to come down and the wooden part to go up, find the average...
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Question: Find the equation of the tangent to the curve $\sqrt{x}+\sqrt{y}=a$ at $\left(\frac{a^{2}}{4}, \frac{a^{2}}{4}\right)$ Solution:...
Read More →Suppose the ends of the coil in the previous problem are
Question: Suppose the ends of the coil in the previous problem are connected to a resistance of $100 \Omega$. Neglecting the resistance of the coil, find the heat produced in the circuit in one minute. Solution:...
Read More →Figure shows a circular coil of N turns and radius a,
Question: Figure shows a circular coil of $\mathrm{N}$ turns and radius a, connected to a battery of emf $\varepsilon$ through a rheostat. The rheostat has a total length $\mathrm{L}$ and resistance $\mathrm{R}$. The resistance of the coil is $\mathrm{r}$. A small circular loop of radius $a^{\prime}$ and resistance $r^{\prime}$ is placed coaxially with the coil. The centre of the loop is at a distance $x$ from the centre of the coil. In the beginning the sliding contact of the rheostat is at the...
Read More →The magnetic field in the cylindrical shown in figure increases at
Question: The magnetic field in the cylindrical shown in figure increases at a constant rate of $20.0 \mathrm{mT} / \mathrm{s}$. Each side of the square loop abcd and defa has a length of $1.00 \mathrm{~cm}$ and a resistance of $4.00 \Omega$. Find the current (magnitude and sense) in the wire ad if (a) the switch $S_{1}$ is closed but $S_{2}$ is open, (b) $S_{1}$ is open but $S_{2}$ is closed, (c) both $S_{1}$ and $S_{2}$ are open and (d) both $S_{1}$ and $S_{2}$ are closed. Solution:...
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